On the range of a two-dimensional conditioned simple random walk

TL;DR

This paper studies the behavior of a two-dimensional simple random walk conditioned on never hitting the origin, revealing its almost recurrent nature and fractal-like range properties.

Contribution

It characterizes the range and visitation patterns of the conditioned walk, showing its almost recurrent behavior and fractal-like structure.

Findings

Each infinite set is visited infinitely often, almost surely.

The proportion of sites visited in large sets is approximately uniform.

There are infinitely many unvisited scaled sets that do not surround the origin.

Abstract

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is "almost recurrent" in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a "large" set, the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset\mathbb{R}^2$ that does not "surround" the origin, we prove that a.s.\ there is an infinite number of $k$'s such that $kG\cap \mathbb{Z}^2$ is unvisited. These results suggest that the range of the conditioned walk has "fractal" behavior.